Numerical Algorithms for Light Transport in Layered Turbid Media and Other Special Mathematical Functions

The study of light transport is important in many applications ranging from neutron transport, gas and plasma dynamics, Brownian motion, and photon transport in biological tissues. In medicine, light can be used to investigate tissue physiology up to several centimeters below the surface without the use of harmful radiation. For example, by spectrally resolving the amount of absorbed near-infrared light, concentrations of both oxygenated and deoxygenated hemoglobin can be derived to measure tissue oxygen saturation and blood volume. On the other hand, measuring the dynamic fluctuations of scattered light from circulating red blood cells can be used to quantify the speed of blood flow. All of these techniques require an appropriate description of light transport through biological tissue that can successfully model the forward transport of light. These models must also be used in inverse models to estimate probed tissue parameters from measured optical signals. Therefore, the successful translation of light based technologies to clinical medicine depend on appropriate physical forward and inverse models that can best describe the underlying tissue structure and experimental system. Because these models are often implemented in computer programs, the accuracy and performance of the translation of these models to numerical algorithms are vital. This dissertation focuses on the intersection of the mathematical theory of light transport and subsequent numerical implementations. An important application of tissue optics is in the study of brain hemodynamics which is limited by the penetration depth of optical signals and the confounding effects of superficial tissue. The use of time-resolved measurements can improve depth sensitivity by selecting for later arriving photons that have a higher probability of probing deeper tissues. The first half of this thesis focuses on developing a time-resolved system and subsequent data analysis procedure for fast analysis of reflectance measurements. An approach using Monte Carlo lookup tables for rapid quantitation of the reduced scattering coefficient within 6-25 % of baseline values was developed. This approach was further improved to include recovery of the absorption and reduced scattering coefficient within 5-15 % that combined the lookup table approach with diffusion based curve fitting. These approaches help reduce the recovered error in optical properties from over 40 % to less than 15 % when considering uncertainties in the measurement's time scale. On the other hand, improved sensitivity to brain tissue can be achieved through more robust data analysis methods. Efficient numerical approaches to solve both the photon diffusion equation and correlation diffusion equation in layered media are given in the steady-state, frequency domain, and time-domain. The developed algorithm was able to simulate the steady-state and time-domain fluence in less than 50 and 500 microseconds which is 3-4 orders of magnitude faster than current approaches allowing for real-time ($